A note on “branch-and-price approach for the multi-skill project scheduling problem”

Abstract

In a recent paper published in Optimization Letters, Montoya et al. (Optim Lett 8:1721-1734, 2014 proposed a branch-and-price approach for a multi-skill project scheduling problem. In that paper, an integer linear programming formulation was first introduced which, unfortunately, has a number of inconsistences. At the interest of mathematical rigor, in this note, we refine such formulation.Keywords Project scheduling · Multi-skilled resources · MILP models In a recent paper by Montoya et al. [2] that was published in Optimization Letters, an integer linear programming formulation was introduced for a multi-skill project scheduling problem. Because a number of inconsistencies could be identified in that formulation, in this note we refine such model at the interest of mathematical rigor. It should be noted that in our opinion, the issues found in the second section of the above mentioned paper do not influence the technical contents of Sections 3-7. In fact, in order to develop their branch-and-price approach, the authors started by developing a column generation formulation which is independent from the aspects we discuss in this note.The relevance of this note arises from the fact that models like the one proposed by Montoya et al. [2] are important for practitioners because, often, they do not have the necessary skills to make full use of integer and combinatorial optimization tools, thus I. Correia (B)

Mixed-integer/linear and constraint programming approaches for activity scheduling in a nuclear research facility

“…A MILP formulation, having similarities with the model proposed by Correia et al (2012), was priorly proposed by Li and Womer (2009) for the MSPSP with minimal and maximal time lags. Discrete-time based formulations have been proposed by Bellenguez-Morineau (2006), Montoya et al (2014) (Correia and Saldanha-da Gama (2015) proposed minor corrections to this model) and Almeida et al (2019), all of them using binary timeindexed step variables. For a given activity i, the binary step variables Z i,t with t lower than the starting time of the activity are all equal to 0, whereas the variables with t equal to or greater than the starting time are all equal to 1.…”

Modeling frameworks for the multi‐skill resource‐constrained project scheduling problem: a theoretical and empirical comparison

“…A third discrete‐time model that we can consider for the MSRCPSP is that by Montoya et al. () with the changes proposed by Correia and Saldanha‐da‐Gama (). We denote this model by .…”

“…We start by revisiting the continuous‐time formulation proposed in that paper as well as the discrete‐time formulation by Montoya et al. () with the changes proposed by Correia and Saldanha‐da‐Gama (). Additionally, we introduce two additional discrete‐time formulations for the problem.…”

“…Montoya et al (2014) proposed a discrete time formulation for the problem addressed by Correia et al (2012). This discrete-time formulation had some inconsistencies that were pointed out by Correia and Saldanha-da-Gama (2015b). More recently, Correia and Saldanha-da-Gama (2015a) introduced a more general modeling framework for an MSRCPSP that includes features of the models proposed by Li and Womer (2009) and Correia et al (2012).…”

“…In this paper, we analyze different integer and mixed-integer formulations for the MSRCPSP studied by Correia et al (2012). We start by revisiting the continuous-time formulation proposed in that paper as well as the discrete-time formulation by Montoya et al (2014) with the changes proposed by Correia and Saldanha-da-Gama (2015b). Additionally, we introduce two additional discrete-time formulations for the problem.…”